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The Hubbard model: A computational perspective
Mingpu Qin, Thomas Schäfer, Sabine Andergassen, Philippe Corboz, Emanuel Gull
TL;DR
The paper reviews how computational methods address the Hubbard model, whose simple local-interaction structure produces rich correlated-electron physics. It synthesizes consensus across numerical approaches while identifying unresolved dynamical, response, and nonequilibrium problems. Recent controlled calculations have established consensus on several phases and mechanisms, including stripe ground states and strong-coupling pseudogaps driven by short-range antiferromagnetic fluctuations.
Problem
The Hubbard model has many competing phases and observables, while exact solutions are limited and precise dynamical and phase-diagram results remain difficult to obtain.
Method
The paper reviews computational methods and defines solving the model through physically interesting observables whose accuracy is comparable to calculations or experiment.
Results
Recent controlled numerical solutions have produced consensus on several aspects, including stripe ground states and a strong-coupling pseudogap caused by short-range antiferromagnetic fluctuations.
Takeaways & Limitations
Multiple complementary methods now resolve important regions of the phase diagram, while tensor networks, dynamical approaches, and vertex-based methods offer prospects for further progress.
Takeaways & Limitations
Reliable generic numerical tools remain largely absent for nonequilibrium and driven Hubbard-model setups.
Abstract
from arXiv · showhide
The Hubbard model is the simplest model of interacting fermions on a lattice and is of similar importance to correlated electron physics as the Ising model is to statistical mechanics or the fruit fly to biomedical science. Despite its simplicity, the model exhibits an incredible wealth of phases, phase transitions, and exotic correlation phenomena. While analytical methods have provided a qualitative description of the model in certain limits, numerical tools have shown impressive progress in achieving quantitative accurate results over the last years. This article gives an introduction to the model, motivates common questions, and illustrates the progress that has been achieved over the last years in revealing various aspects of the correlation physics of the model.
1. Introduction
The Hubbard model is a simple lattice model that isolates local electron correlations while supporting diverse phases and phenomena. This review surveys computational progress, defines practical standards for solving the model, and emphasizes multi-method benchmarks.
- Model definition: The Hubbard model describes interacting lattice fermions through hopping t and local interaction strength U.Its formulation emphasizes correlation physics from local interactions in a single orbital.
- Motivation: Despite excluding non-local interactions, band structure, and multi-orbital effects, the model remains a powerful testbed for correlated-electron physics and quantum simulation.Its simplicity also makes it an appealing target for analytical methods and early quantum simulators.
- Computational motivation: Analytical methods alone cannot describe the model’s rich metallic, insulating, magnetic, superconducting, and charge-ordered physics with the desired accuracy.This limitation motivated a wide range of numerical tools based on different approximations and approaches.
- Scope: The review focuses primarily on the single-band square-lattice model from weak through strong coupling, with connections to the t-J model and three dimensions.Other geometries, attractive interactions, multi-orbital systems, non-local interactions, and driven systems receive only brief discussion.
- Standards for solving: “Solving” the Hubbard model means obtaining a physically interesting observable with accuracy comparable to or better than comparison calculations or experiment.What counts as solving depends on the observable and parameter regime, and changes as numerical precision improves.
- Benchmarking: Multi-method consensus studies are valuable because controlled extrapolations can be difficult and uncontrolled methods may still be accurate or especially suitable in particular regimes.Benchmark projects compare results obtained with different underlying approximations at the same parameters.
2. Phase diagram and phase boundaries
The two-dimensional Hubbard-model phase diagram has broad agreement on phases and approximate locations but remains unsettled in its precise boundaries and shapes. Competing orders, crossovers, finite-size effects, and approximation errors make detailed determination difficult.
- Open phase diagram: The intermediate-to-strong-coupling phase diagram remains hotly debated, especially because competing phases can have large and distinct regions of parameter space.Weak-coupling ground-state behavior is comparatively well understood.
- Phase boundaries: Divergent correlation lengths at continuous transitions complicate finite-size scaling and hinder precise phase-boundary determination.Broad crossovers can separate physical regimes rather than sharp or first-order transitions.
- Methodological complications: Approximate methods can break symmetries and overemphasize ordered phases, while finite-size methods may predict spurious finite-temperature magnetic order.Such order disappears only slowly as system size becomes very large.
- Current consensus: A broad consensus exists on the phases and their approximate locations, but the precise shapes of transition lines are often unknown.Consequently, visual phase diagrams may reflect a particular approximation or comparisons with simple theories and experiments.
- Intertwined orders: Many orders and phenomena are intertwined because they coexist or lie close in energy or parameter space, complicating identification of individual phases.Additional proposed phases include nematic phases.
3. The Hubbard model at half filling
At half filling, computational methods have established detailed results across interaction and temperature regimes, including insulating and antiferromagnetic ground states. Dynamical quantities and finite-temperature behavior remain technically challenging, especially near divergent correlation lengths.
- Methods: The fermion sign problem is absent at half filling on bipartite lattices, enabling efficient large-lattice QMC and thermodynamic-limit extrapolations.Other numerical methods also provide precise results for many physical quantities.
- Static observables: Half-filled studies compute energies, entropies, spin and charge correlations, order parameters, double occupancies, and spectral quantities using complementary numerical methods.Finite-size, finite-temperature, and thermodynamic-limit results have been reported across interaction regimes.
- Two-dimensional finite-temperature physics: In weak coupling, the magnetic correlation length ξ grows exponentially as temperature decreases, contributing to magnetic-order destruction and weak-coupling pseudogap formation.These long-range fluctuations are associated with the Mermin-Wagner theorem.
- Dynamical quantities: Full frequency-and-momentum spectral resolution remains an outstanding problem because cluster methods provide limited momentum information and periodization choices strongly affect detailed results.Tensor-network methods offer complementary dynamical calculations but have so far been applied to selected geometries and regimes.
- Ground state and gap: The half-filled square-lattice ground state with t′ = 0 is insulating with a charge gap for every U, and antiferromagnetic Néel order persists for all interaction strengths.The strong-coupling gap is Mott-like, while weak-coupling gap formation is associated with competing Mott and Slater interpretations.
- Nature of the gap: A crossover near U/t ≈4 separates weak- and strong-coupling gap regimes without an observed phase transition.Weak-coupling energy trends support a Slater mechanism, while strong coupling is described by an effective Heisenberg model with J = 4t^2/U.
- Temperature evolution: Cooling produces crossovers from incoherent high-temperature behavior to coherent metallic quasiparticles and then an insulating antiferromagnetic pseudogap.The inverse quasiparticle lifetime has a temperature-dependent minimum as low-temperature fluctuation rates increase.
- Three dimensions: In three dimensions, the half-filled model transitions from a high-temperature paramagnet to a low-temperature antiferromagnetic insulator, with maximum TN/t ∼0.35 near U/t ∼8.The transition temperature decreases rapidly at weak and strong coupling.
4. The doped 2D Hubbard model at weak coupling
At weak coupling, magnetic fluctuations generate rich behavior in the doped 2D Hubbard model, including incommensurate order, d-wave pairing, and pseudogap formation. Numerical and analytical approaches connect these phenomena to magnetic correlations, Fermi-surface topology, and thermal length scales.
- Magnetic properties: Incommensurate magnetic order appears away from half-filling, supported by mean-field, hole-density expansions, and fRG susceptibility divergences.
- Superconductivity: Pairing is fluctuation-driven and difficult for mean-field theory, while computational approaches establish d-wave superconductivity at weak and moderate coupling.
- Superconductivity: DMF2RG confirms robust strong-coupling d-wave pairing driven by magnetic correlations near the antiferromagnetic regime.
- Pseudogap: The pseudogap is a momentum-dependent suppression of spectral weight near the Fermi energy induced at weak to intermediate coupling by long-range antiferromagnetic correlations.It opens when the correlation length exceeds the thermal de Broglie wavelength, ξ ≫vF /(πT).
- Pseudogap: Second-order self-energy analysis predicts a gap near hot spots, opening first at the antinode and spreading toward the node as temperature decreases.
- Fermi-surface effects: Weak-coupling quasiparticle coherence depends on Fermi-surface topology, with suppression at hot spots for hole-like Fermi surfaces.
5.1. Competition of low-energy ground states: uniform vs stripe states
The doped 2D Hubbard model contains closely competing uniform d-wave and stripe states, making the ground state difficult to determine. Multi-method studies establish stripes in important strong-coupling regimes, while intermediate coupling, superconducting coexistence, and quantitative phase boundaries remain unsettled.
- Competition of low-energy ground states: Uniform d-wave superconductivity and inhomogeneous stripe states can lie very close in energy, producing long-standing ground-state competition.
- Competition at intermediate U/t: At U/t = 8 and δ = 1/8, DMRG, CP-AFQMC, DMET, and iPEPS reach consensus on a charge-period-8 stripe without coexisting d-wave superconductivity.
- Competition at intermediate U/t: For the period-8 stripe, four methods estimate an energy of −0.767±0.004t, while the uniform d-wave state is higher by ≈0.01t.
- Doping dependence: Stripe periods generally decrease with increasing doping, while superconducting coexistence and phase separation predictions vary across methods and remain incompletely confirmed.
- Uniform versus stripe states: Uniform d-wave stability depends on doping, interaction, and longer-range terms, with reported stable regions differing across computational approaches.
- Competition at intermediate U/t: At U/t = 4, stripe formation is weaker than at strong coupling, and the crossover between uniform and stripe states remains unresolved.
- Stripes at finite t′: Negative t′ favors shorter stripe periods, while sufficiently large positive t′ stabilizes the uniform d-wave state over stripes.
- Stripes at finite t′: Coexistence or absence of superconductivity in stripe states remains an important open issue, including the role of spin-order range and phase shifts.
5.2. Pseudogap
The pseudogap is studied using cluster methods that capture short-range correlations and reveal momentum-selective behavior. Its organizing principles and microscopic origin remain subject to competing interpretations.
- Cluster methods such as DCA and CDMFT capture short-range correlations and provide access to spectral functions.
- Lowering T below T* suppresses the antinodal density of states while leaving the nodal region less affected.
- Organizing principles: Systematic DCA studies describe the pseudogap as a doping-driven Mott transition, with a gap opening near the antinode while the node remains metallic.
- Organizing principles: A 2×2 CDMFT analysis identifies a Widom line where thermodynamic and dynamic criteria for T* collapse onto thermodynamic anomalies from the Mott critical endpoint.
- Open questions: The influence of nesting and the van Hove singularity at strong coupling remains unresolved because numerical studies report conflicting effects.
- Origin: Fluctuation diagnostics finds a clear Q = (π, π) peak and dominant zero-frequency contribution in the spin channel of the pseudogap self-energy.
5.3. Superconductivity
The Hubbard model supports superconductivity across relevant parameter regimes, with numerical methods determining its symmetry, phase boundary, and state properties. Antiferromagnetic fluctuations have been identified as a pairing contribution, while particle-particle processes can suppress Tc.
- Superconducting phase boundaries are located either by diverging susceptibility on cooling or by disappearance of the order parameter on heating.
- Normal-state QMC and DCA studies identify dx2−y2 as the most likely pairing symmetry for moderate doping and determine favorable parameter regimes.
- Cluster DMFT simulations analyze superconducting-state energetics, spectral functions, gap functions, response functions, and phase boundaries.
- Finite-size effects remain, but generic energetic and response-function trends coincide across methods and with observations on cuprate materials.
- Pairing mechanism: Several studies identify antiferromagnetic fluctuations as pairing glue, while particle-particle scattering processes are identified as a main suppressor of Tc.
5.4. Bad metal
At high temperature and intermediate doping, the Hubbard model is metallic but has transport properties inconsistent with Fermi-liquid behavior. Reliable calculations remain unavailable in an intermediate-temperature regime.
- At high temperature and intermediate doping, the model is metallic but its transport properties are inconsistent with Fermi-liquid behavior.
- Resistivity calculations are difficult because analytic continuation is unreliable at high temperature and real-frequency DMFT formulations neglect vertex corrections.
- The very-high-temperature regime is well understood, but no reliable calculations currently exist in the intermediate-temperature regime.
- Modern real-frequency DiagMC methods are proposed as a possible resolution to the intermediate-temperature transport problem.
6. Towards the simulation of experimental probes
Computational methods increasingly connect Hubbard-model calculations with experimental probes, while spectral reconstruction and momentum-resolved magnetic response remain challenging. Simulations reproduce several experimentally observed signatures across spectroscopies and response functions.
- Single- and two-particle excitations are less well known at intermediate-to-strong coupling than the model’s energetics and phase diagram.
- Spectral functions: Analytic continuation connects imaginary-axis calculations to spectral functions, but its ill-conditioned kernel exponentially amplifies noise and uncertainty.
- Spectral functions: Matsubara self-energy fits can provide quasiparticle weights, gap sizes, and metal-to-insulator transition locations without analytic continuation.
- Transport response: Simulated interplane conductivities show pseudogap signatures with temperature and doping that are consistent with cuprate experiments.
- Raman response: Simulated Raman spectra show a two-magnon peak in the insulator, doping-related pseudogap signatures, and temperature and doping evolution generally consistent with experiment.
- Magnetic response: Full momentum- and energy-dependent magnetic susceptibility remains an open problem because available lattice and quantum-cluster methods lack unbiased momentum resolution.
7. Generalizations and extensions
Generalizations to other lattices and interaction signs reveal additional ordered, superconducting, and spin-liquid regimes, while several phase boundaries remain debated.
- Honeycomb lattice: On the honeycomb lattice, a direct Dirac semi-metal to antiferromagnetic Mott-insulator transition occurs at U_c/t ≈3.8, with no intermediate spin liquid.The transition belongs to the Gross-Neveu-Yukawa universality class.
- Honeycomb lattice: At one-quarter doping on the honeycomb lattice, several methods find chiral d+id superconducting order.
- Triangular lattice: Triangular-lattice studies agree on 120-degree magnetic order at half filling in the Heisenberg limit, but disagree over an intermediate chiral spin liquid.A recent DMRG study found such a phase, whereas VMC did not except with sufficiently large t′.
- Kagome lattice: The Kagome lattice shows no magnetic order at half filling in the strongly interacting limit, with growing consensus for a quantum spin-liquid ground state.Cluster DMFT found a Mott-transition coupling U_c/t ∼8.22.
- Attractive model: For the attractive Hubbard model, QMC remains sign-problem free, enabling accurate large-system low-temperature calculations; away from half filling, onsite pairing survives.At half filling, onsite s-wave pairing and charge-density-wave order coexist, while finite-temperature behavior includes a Kosterlitz-Thouless transition away from half filling.
7.3. Multi-orbital models
The review extends the single-band Hubbard framework by adding orbitals, non-local interactions, and nonequilibrium driving, while emphasizing material specificity and missing generic tools.
- Multi-orbital models: A bottom-up route toward realistic electronic-structure Hamiltonians adds orbitals and interactions progressively, beginning with local multi-orbital Hubbard models.The number of models and parameter choices quickly becomes material specific.
- Multi-orbital models: Three-orbital Emery models describe cuprate-motivated physics, while oxide-perovskite generalizations include Slater-Kanamori interactions with parameters U, U′, and J.These models have been studied most often using single-site multi-orbital DMFT and have produced the concept of Hunds’ metals.
- Non-local interactions: The extended Hubbard model adds a repulsive nearest-neighbor density interaction V, driving charge order as V increases and temperature decreases.Its behavior includes competition or coexistence among charge order, correlation, antiferromagnetism, metallicity, and Mott insulation.
- Non-local interactions: Extended-Hubbard interactions are treated explicitly in DCA or through diagrammatic expansions around DMFT in the dual-boson approach.
- Nonequilibrium and driven systems: Reliable and generic numerical tools for real-time nonequilibrium Hubbard setups are mostly absent, despite existing single-site, small-cluster, and Floquet studies.
8. Conclusions and perspectives
Recent computational advances have produced controlled solutions and consensus on several Hubbard-model phenomena, but phase boundaries, competing orders, and broadly applicable numerical methods remain open challenges.
- Conclusions: Controlled numerical solutions are now available across various regions of the phase diagram, yielding consensus on several aspects of the Hubbard model.
- Conclusions: Stripe states are widely accepted as ground states over an extended range of doping, interaction strength, and t′.
- Conclusions: The pseudogap at strong coupling is associated with strong short-range antiferromagnetic fluctuations.
- Open challenges: Open challenges include locating phase boundaries, clarifying low-doping phase separation, and understanding superconductivity in partially filled stripes.The parameter region with d-wave superconductivity remains especially uncertain.
- Perspectives: Tensor-network methods have reached finite temperature in two dimensions, while dynamical variational Monte Carlo and real-frequency vertex approaches are advancing dynamical calculations.These developments support prospects for addressing outstanding issues.
- Perspectives: Finite-temperature methods have advanced through large-cluster DMFT, more accurate DMFT extensions, and continuous-time and diagrammatic Monte Carlo techniques.The latter methods are described as orders of magnitude more powerful than previous techniques.
- Open challenges: Developing generally applicable numerical techniques that overcome existing shortcomings remains the most urgent need.The review traces most advances over the last 15 years to algorithmic and numerical progress.
DISCLOSURE STATEMENT
The authors report no affiliations, memberships, funding, or financial holdings that might affect the review’s objectivity.
- The authors declare no affiliations, memberships, funding, or financial holdings perceived as affecting objectivity.
LITERATURE CITED
The literature-cited section compiles references relevant to the Hubbard model and correlated-electron physics across review articles, journal papers, reports, and preprints. Entries include publication metadata such as authors, venues, years, identifiers, and URLs.
- Reference types: The bibliography includes review articles in venues such as Annual Reviews, Reviews of Modern Physics, and Advances in Physics.Examples include Agterberg et al. (2020), Aoki et al. (2014), and Anderson (1997).
- Reference metadata: Entries provide persistent identifiers or web addresses for locating the cited works.The bibliography lists DOI links, APS URLs, publisher URLs, and arXiv identifiers.
- Reference types: Many cited works appear in Physical Review Letters and Physical Review B, with publication years spanning several decades.The entries include papers dated from 1989 through 2021.
- Reference types: The references also include Science reports and arXiv e-prints alongside journal publications.Science entries include Uehlinger et al. (2013) and a 2019 report; arXiv entries include Qin et al. (2021) and Arute et al. (2020).